# Properties

 Label 1216.2.h.c Level $1216$ Weight $2$ Character orbit 1216.h Analytic conductor $9.710$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 304) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} + 4 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} + 4 q^{9} + 2 \beta_{2} q^{11} -\beta_{3} q^{13} + 3 q^{17} + ( -\beta_{1} + 2 \beta_{2} ) q^{19} -\beta_{3} q^{21} -3 \beta_{2} q^{23} -5 q^{25} + \beta_{1} q^{27} -\beta_{3} q^{29} + 2 \beta_{1} q^{31} -2 \beta_{3} q^{33} + 2 \beta_{3} q^{37} + 7 \beta_{2} q^{39} + 2 \beta_{3} q^{41} + 2 \beta_{2} q^{47} + 4 q^{49} + 3 \beta_{1} q^{51} + \beta_{3} q^{53} + ( -7 - 2 \beta_{3} ) q^{57} + 3 \beta_{1} q^{59} + 8 q^{61} + 4 \beta_{2} q^{63} + \beta_{1} q^{67} + 3 \beta_{3} q^{69} + 6 \beta_{1} q^{71} -11 q^{73} -5 \beta_{1} q^{75} -6 q^{77} -2 \beta_{1} q^{79} -5 q^{81} + 4 \beta_{2} q^{83} + 7 \beta_{2} q^{87} + 4 \beta_{3} q^{89} -3 \beta_{1} q^{91} + 14 q^{93} + 2 \beta_{3} q^{97} + 8 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 16q^{9} + O(q^{10})$$ $$4q + 16q^{9} + 12q^{17} - 20q^{25} + 16q^{49} - 28q^{57} + 32q^{61} - 44q^{73} - 24q^{77} - 20q^{81} + 56q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/7$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{2} + 7$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 14 \nu$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$7 \beta_{2} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1215.1
 1.32288 − 2.29129i 1.32288 + 2.29129i −1.32288 + 2.29129i −1.32288 − 2.29129i
0 −2.64575 0 0 0 1.73205i 0 4.00000 0
1215.2 0 −2.64575 0 0 0 1.73205i 0 4.00000 0
1215.3 0 2.64575 0 0 0 1.73205i 0 4.00000 0
1215.4 0 2.64575 0 0 0 1.73205i 0 4.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.h.c 4
4.b odd 2 1 inner 1216.2.h.c 4
8.b even 2 1 304.2.h.b 4
8.d odd 2 1 304.2.h.b 4
19.b odd 2 1 inner 1216.2.h.c 4
24.f even 2 1 2736.2.k.k 4
24.h odd 2 1 2736.2.k.k 4
76.d even 2 1 inner 1216.2.h.c 4
152.b even 2 1 304.2.h.b 4
152.g odd 2 1 304.2.h.b 4
456.l odd 2 1 2736.2.k.k 4
456.p even 2 1 2736.2.k.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.b 4 8.b even 2 1
304.2.h.b 4 8.d odd 2 1
304.2.h.b 4 152.b even 2 1
304.2.h.b 4 152.g odd 2 1
1216.2.h.c 4 1.a even 1 1 trivial
1216.2.h.c 4 4.b odd 2 1 inner
1216.2.h.c 4 19.b odd 2 1 inner
1216.2.h.c 4 76.d even 2 1 inner
2736.2.k.k 4 24.f even 2 1
2736.2.k.k 4 24.h odd 2 1
2736.2.k.k 4 456.l odd 2 1
2736.2.k.k 4 456.p even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1216, [\chi])$$:

 $$T_{3}^{2} - 7$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -7 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 3 + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$( 21 + T^{2} )^{2}$$
$17$ $$( -3 + T )^{4}$$
$19$ $$361 + 10 T^{2} + T^{4}$$
$23$ $$( 27 + T^{2} )^{2}$$
$29$ $$( 21 + T^{2} )^{2}$$
$31$ $$( -28 + T^{2} )^{2}$$
$37$ $$( 84 + T^{2} )^{2}$$
$41$ $$( 84 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$( 12 + T^{2} )^{2}$$
$53$ $$( 21 + T^{2} )^{2}$$
$59$ $$( -63 + T^{2} )^{2}$$
$61$ $$( -8 + T )^{4}$$
$67$ $$( -7 + T^{2} )^{2}$$
$71$ $$( -252 + T^{2} )^{2}$$
$73$ $$( 11 + T )^{4}$$
$79$ $$( -28 + T^{2} )^{2}$$
$83$ $$( 48 + T^{2} )^{2}$$
$89$ $$( 336 + T^{2} )^{2}$$
$97$ $$( 84 + T^{2} )^{2}$$